Error analysis based on inverse modified differential equations for discovery of dynamics using linear multistep methods and deep learning

Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics via linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the inverse modified differential equations (IMDE) of linear multistep methods and show that the learning model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target dynamical system is bounded by the sum of the LMM discretization error and the learning loss. Several numerical experiments are performed to verify the theoretical analysis.